/*
 * Licensed to the Apache Software Foundation (ASF) under one or more
 * contributor license agreements.  See the NOTICE file distributed with
 * this work for additional information regarding copyright ownership.
 * The ASF licenses this file to You under the Apache License, Version 2.0
 * (the "License"); you may not use this file except in compliance with
 * the License.  You may obtain a copy of the License at
 * 
 *      http://www.apache.org/licenses/LICENSE-2.0
 * 
 * Unless required by applicable law or agreed to in writing, software
 * distributed under the License is distributed on an "AS IS" BASIS,
 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
 * See the License for the specific language governing permissions and
 * limitations under the License.
 */
package com.landawn.abacus.util;

import java.math.BigInteger;

/**
 * <p>
 * Note: it's copied from Apache Commons Lang developed at The Apache Software Foundation (http://www.apache.org/), or
 * under the Apache License 2.0. The methods copied from other products/frameworks may be modified in this class.
 * </p>
 * 
 * <p>
 * <code>Fraction</code> is a <code>Number</code> implementation that stores fractions accurately.
 * </p>
 * 
 * <p>
 * This class is immutable, and interoperable with most methods that accept a <code>Number</code>.
 * </p>
 * 
 * <p>
 * Note that this class is intended for common use cases, it is <i>int</i> based and thus suffers from various overflow
 * issues. For a BigInteger based equivalent, please see the Commons Math BigFraction class.
 * </p>
 *
 * @version $Id: Fraction.java 1583482 2014-03-31 22:54:57Z niallp $
 * @since 2.0
 */
@com.landawn.abacus.annotation.Immutable
public final class Fraction extends Number implements Comparable<Fraction>, Immutable {

    /**
     * Required for serialization support. Lang version 2.0.
     * 
     * @see java.io.Serializable
     */
    private static final long serialVersionUID = 65382027393090L;

    /**
     * <code>Fraction</code> representation of 0.
     */
    public static final Fraction ZERO = new Fraction(0, 1);
    /**
     * <code>Fraction</code> representation of 1.
     */
    public static final Fraction ONE = new Fraction(1, 1);
    /**
     * <code>Fraction</code> representation of 1/2.
     */
    public static final Fraction ONE_HALF = new Fraction(1, 2);
    /**
     * <code>Fraction</code> representation of 1/3.
     */
    public static final Fraction ONE_THIRD = new Fraction(1, 3);
    /**
     * <code>Fraction</code> representation of 2/3.
     */
    public static final Fraction TWO_THIRDS = new Fraction(2, 3);
    /**
     * <code>Fraction</code> representation of 1/4.
     */
    public static final Fraction ONE_QUARTER = new Fraction(1, 4);
    /**
     * <code>Fraction</code> representation of 2/4.
     */
    public static final Fraction TWO_QUARTERS = new Fraction(2, 4);
    /**
     * <code>Fraction</code> representation of 3/4.
     */
    public static final Fraction THREE_QUARTERS = new Fraction(3, 4);
    /**
     * <code>Fraction</code> representation of 1/5.
     */
    public static final Fraction ONE_FIFTH = new Fraction(1, 5);
    /**
     * <code>Fraction</code> representation of 2/5.
     */
    public static final Fraction TWO_FIFTHS = new Fraction(2, 5);
    /**
     * <code>Fraction</code> representation of 3/5.
     */
    public static final Fraction THREE_FIFTHS = new Fraction(3, 5);
    /**
     * <code>Fraction</code> representation of 4/5.
     */
    public static final Fraction FOUR_FIFTHS = new Fraction(4, 5);

    /**
     * The numerator number part of the fraction (the three in three sevenths).
     */
    private final int numerator;
    /**
     * The denominator number part of the fraction (the seven in three sevenths).
     */
    private final int denominator;

    /**
     * Cached output hashCode (class is immutable).
     */
    private transient int hashCode = 0;
    /**
     * Cached output toString (class is immutable).
     */
    private transient String toString = null;
    /**
     * Cached output toProperString (class is immutable).
     */
    private transient String toProperString = null;

    /**
     * <p>
     * Constructs a <code>Fraction</code> instance with the 2 parts of a fraction Y/Z.
     * </p>
     * 
     * @param numerator
     *            the numerator, for example the three in 'three sevenths'
     * @param denominator
     *            the denominator, for example the seven in 'three sevenths'
     */
    private Fraction(final int numerator, final int denominator) {
        super();
        this.numerator = numerator;
        this.denominator = denominator;
    }

    /**
     *
     * @param numerator
     * @param denominator
     * @return
     * @see #of(int, int, boolean)
     */
    public static Fraction of(int numerator, int denominator) {
        return of(numerator, denominator, false);
    }

    /**
     * <p>
     * Creates a <code>Fraction</code> instance with the 2 parts of a fraction Y/Z.
     * </p>
     * 
     * <p>
     * Any negative signs are resolved to be on the numerator.
     * </p>
     * 
     * @param numerator
     *            the numerator, for example the three in 'three sevenths'
     * @param denominator
     *            the denominator, for example the seven in 'three sevenths'
     * @param reduce
     *            if it's true, reduce <code>Fraction</code> instance with the 2 parts of a fraction Y/Z. For example,
     *            if the input parameters represent 2/4, then the created fraction will be 1/2.
     * @return a new fraction instance
     * @throws ArithmeticException
     *             if the denominator is <code>zero</code> or the denominator is {@code negative} and the numerator is
     *             {@code Integer#MIN_VALUE}
     */
    public static Fraction of(int numerator, int denominator, boolean reduce) {

        if (denominator == 0) {
            throw new ArithmeticException("The denominator must not be zero");
        }

        if (reduce) {
            // allow 2^k/-2^31 as a valid fraction (where k>0)
            if (denominator == Integer.MIN_VALUE && (numerator & 1) == 0) {
                numerator /= 2;
                denominator /= 2;
            }
        }

        if (denominator < 0) {
            if (numerator == Integer.MIN_VALUE || denominator == Integer.MIN_VALUE) {
                throw new ArithmeticException("overflow: can't negate");
            }
            numerator = -numerator;
            denominator = -denominator;
        }

        if (reduce) {
            if (numerator == 0) {
                return ZERO; // normalize zero.
            }
            // simplify fraction.
            final int gcd = greatestCommonDivisor(numerator, denominator);
            numerator /= gcd;
            denominator /= gcd;
            return new Fraction(numerator, denominator);
        } else {
            return new Fraction(numerator, denominator);
        }
    }

    /**
     *
     * @param whole
     * @param numerator
     * @param denominator
     * @return
     * @see {{@link #of(int, int, int, boolean)}
     */
    public static Fraction of(final int whole, final int numerator, final int denominator) {
        return of(whole, numerator, denominator, false);
    }

    /**
     * <p>
     * Creates a <code>Fraction</code> instance with the 3 parts of a fraction X Y/Z.
     * </p>
     * 
     * <p>
     * The negative sign must be passed in on the whole number part.
     * </p>
     *
     * @param whole the whole number, for example the one in 'one and three sevenths'
     * @param numerator the numerator, for example the three in 'one and three sevenths'
     * @param denominator the denominator, for example the seven in 'one and three sevenths'
     * @param reduce
     * @return a new fraction instance
     * @throws ArithmeticException             if the resulting numerator exceeds <code>Integer.MAX_VALUE</code>
     */
    public static Fraction of(final int whole, final int numerator, final int denominator, boolean reduce) {

        if (denominator == 0) {
            throw new ArithmeticException("The denominator must not be zero");
        }
        if (denominator < 0) {
            throw new ArithmeticException("The denominator must not be negative");
        }
        if (numerator < 0) {
            throw new ArithmeticException("The numerator must not be negative");
        }
        long numeratorValue;
        if (whole < 0) {
            numeratorValue = whole * (long) denominator - numerator;
        } else {
            numeratorValue = whole * (long) denominator + numerator;
        }
        if (numeratorValue < Integer.MIN_VALUE || numeratorValue > Integer.MAX_VALUE) {
            throw new ArithmeticException("Numerator too large to represent as an Integer.");
        }

        return of((int) numeratorValue, denominator, reduce);
    }

    /**
     * <p>
     * Creates a <code>Fraction</code> instance from a <code>double</code> value.
     * </p>
     * 
     * <p>
     * This method uses the <a href="http://archives.math.utk.edu/articles/atuyl/confrac/"> continued fraction
     * algorithm</a>, computing a maximum of 25 convergents and bounding the denominator by 10,000.
     * </p>
     *
     * @param value the double value to convert
     * @return a new fraction instance that is close to the value
     * @throws ArithmeticException             if the the algorithm does not converge
     */
    public static Fraction of(double value) {
        final int sign = value < 0 ? -1 : 1;
        value = Math.abs(value);
        if (value > Integer.MAX_VALUE || Double.isNaN(value)) {
            throw new ArithmeticException("The value must not be greater than Integer.MAX_VALUE or NaN");
        }
        final int wholeNumber = (int) value;
        value -= wholeNumber;

        int numer0 = 0; // the pre-previous
        int denom0 = 1; // the pre-previous
        int numer1 = 1; // the previous
        int denom1 = 0; // the previous
        int numer2 = 0; // the current, setup in calculation
        int denom2 = 0; // the current, setup in calculation
        int a1 = (int) value;
        int a2 = 0;
        double x1 = 1;
        double x2 = 0;
        double y1 = value - a1;
        double y2 = 0;
        double delta1, delta2 = Double.MAX_VALUE;
        double fraction;
        int i = 1;
        //        System.out.println("---");
        do {
            delta1 = delta2;
            a2 = (int) (x1 / y1);
            x2 = y1;
            y2 = x1 - a2 * y1;
            numer2 = a1 * numer1 + numer0;
            denom2 = a1 * denom1 + denom0;
            fraction = (double) numer2 / (double) denom2;
            delta2 = Math.abs(value - fraction);
            //            System.out.println(numer2 + " " + denom2 + " " + fraction + " " + delta2 + " " + y1);
            a1 = a2;
            x1 = x2;
            y1 = y2;
            numer0 = numer1;
            denom0 = denom1;
            numer1 = numer2;
            denom1 = denom2;
            i++;
            //            System.out.println(">>" + delta1 +" "+ delta2+" "+(delta1 > delta2)+" "+i+" "+denom2);
        } while (delta1 > delta2 && denom2 <= 10000 && denom2 > 0 && i < 25);
        if (i == 25) {
            throw new ArithmeticException("Unable to convert double to fraction");
        }
        return of((numer0 + wholeNumber * denom0) * sign, denom0, true);
    }

    /**
     * <p>
     * Creates a Fraction from a <code>String</code>.
     * </p>
     * 
     * <p>
     * The formats accepted are:
     * </p>
     * 
     * <ol>
     * <li><code>double</code> String containing a dot</li>
     * <li>'X Y/Z'</li>
     * <li>'Y/Z'</li>
     * <li>'X' (a simple whole number)</li>
     * </ol>
     * <p>
     * and a .
     * </p>
     * 
     * @param str
     *            the string to parse, must not be <code>null</code>
     * @return
     * @throws IllegalArgumentException
     *             if the string is <code>null</code>
     * @throws NumberFormatException
     *             if the number format is invalid
     */
    public static Fraction of(String str) {
        if (str == null) {
            throw new IllegalArgumentException("The string must not be null");
        }
        // parse double format
        int pos = str.indexOf('.');
        if (pos >= 0) {
            return of(Double.parseDouble(str));
        }

        // parse X Y/Z format
        pos = str.indexOf(' ');
        if (pos > 0) {
            final int whole = Integer.parseInt(str.substring(0, pos));
            str = str.substring(pos + 1);
            pos = str.indexOf('/');
            if (pos < 0) {
                throw new NumberFormatException("The fraction could not be parsed as the format X Y/Z");
            } else {
                final int numer = Integer.parseInt(str.substring(0, pos));
                final int denom = Integer.parseInt(str.substring(pos + 1));
                return of(whole, numer, denom);
            }
        }

        // parse Y/Z format
        pos = str.indexOf('/');
        if (pos < 0) {
            // simple whole number
            return of(Integer.parseInt(str), 1);
        } else {
            final int numer = Integer.parseInt(str.substring(0, pos));
            final int denom = Integer.parseInt(str.substring(pos + 1));
            return of(numer, denom);
        }
    }

    // Accessors
    //-------------------------------------------------------------------

    /**
     * <p>
     * Gets the numerator part of the fraction.
     * </p>
     * 
     * <p>
     * This method may return a value greater than the denominator, an improper fraction, such as the seven in 7/4.
     * </p>
     * 
     * @return
     * @deprecated replaced by {@code numerator}
     */
    @Deprecated
    public int getNumerator() {
        return numerator;
    }

    public int numerator() {
        return numerator;
    }

    /**
     * <p>
     * Gets the denominator part of the fraction.
     * </p>
     * 
     * @return
     * @deprecated replaced by {@code denominator}
     */
    @Deprecated
    public int getDenominator() {
        return denominator;
    }

    public int denominator() {
        return denominator;
    }

    /**
     * <p>
     * Gets the proper numerator, always positive.
     * </p>
     * 
     * <p>
     * An improper fraction 7/4 can be resolved into a proper one, 1 3/4. This method returns the 3 from the proper
     * fraction.
     * </p>
     * 
     * <p>
     * If the fraction is negative such as -7/4, it can be resolved into -1 3/4, so this method returns the positive
     * proper numerator, 3.
     * </p>
     * 
     * @return
     * @deprecated replaced by {@code properNumerator}
     */
    @Deprecated
    public int getProperNumerator() {
        return Math.abs(numerator % denominator);
    }

    public int properNumerator() {
        return Math.abs(numerator % denominator);
    }

    /**
     * <p>
     * Gets the proper whole part of the fraction.
     * </p>
     * 
     * <p>
     * An improper fraction 7/4 can be resolved into a proper one, 1 3/4. This method returns the 1 from the proper
     * fraction.
     * </p>
     * 
     * <p>
     * If the fraction is negative such as -7/4, it can be resolved into -1 3/4, so this method returns the positive
     * whole part -1.
     * </p>
     * 
     * @return
     * @deprecated replaced by {@code properWhole}
     */
    @Deprecated
    public int getProperWhole() {
        return numerator / denominator;
    }

    public int properWhole() {
        return numerator / denominator;
    }

    // Number methods
    //-------------------------------------------------------------------

    /**
     * <p>
     * Gets the fraction as an <code>int</code>. This returns the whole number part of the fraction.
     * </p>
     * 
     * @return
     */
    @Override
    public int intValue() {
        return numerator / denominator;
    }

    /**
     * <p>
     * Gets the fraction as a <code>long</code>. This returns the whole number part of the fraction.
     * </p>
     * 
     * @return
     */
    @Override
    public long longValue() {
        return (long) numerator / denominator;
    }

    /**
     * <p>
     * Gets the fraction as a <code>float</code>. This calculates the fraction as the numerator divided by denominator.
     * </p>
     * 
     * @return
     */
    @Override
    public float floatValue() {
        return (float) numerator / (float) denominator;
    }

    /**
     * <p>
     * Gets the fraction as a <code>double</code>. This calculates the fraction as the numerator divided by denominator.
     * </p>
     * 
     * @return
     */
    @Override
    public double doubleValue() {
        return (double) numerator / (double) denominator;
    }

    // Calculations
    //-------------------------------------------------------------------

    /**
     * <p>
     * Reduce the fraction to the smallest values for the numerator and denominator, returning the result.
     * </p>
     * 
     * <p>
     * For example, if this fraction represents 2/4, then the result will be 1/2.
     * </p>
     * 
     * @return a new reduced fraction instance, or this if no simplification possible
     */
    public Fraction reduce() {
        if (numerator == 0) {
            return equals(ZERO) ? this : ZERO;
        }
        final int gcd = greatestCommonDivisor(Math.abs(numerator), denominator);
        if (gcd == 1) {
            return this;
        }
        return Fraction.of(numerator / gcd, denominator / gcd);
    }

    /**
     * <p>
     * Gets a fraction that is the inverse (1/fraction) of this one.
     * </p>
     * 
     * <p>
     * The returned fraction is not reduced.
     * </p>
     * 
     * @return a new fraction instance with the numerator and denominator inverted.
     * @throws ArithmeticException
     *             if the fraction represents zero.
     */
    public Fraction invert() {
        if (numerator == 0) {
            throw new ArithmeticException("Unable to invert zero.");
        }
        if (numerator == Integer.MIN_VALUE) {
            throw new ArithmeticException("overflow: can't negate numerator");
        }
        if (numerator < 0) {
            return new Fraction(-denominator, -numerator);
        } else {
            return new Fraction(denominator, numerator);
        }
    }

    /**
     * <p>
     * Gets a fraction that is the negative (-fraction) of this one.
     * </p>
     * 
     * <p>
     * The returned fraction is not reduced.
     * </p>
     * 
     * @return a new fraction instance with the opposite signed numerator
     */
    public Fraction negate() {
        // the positive range is one smaller than the negative range of an int.
        if (numerator == Integer.MIN_VALUE) {
            throw new ArithmeticException("overflow: too large to negate");
        }
        return new Fraction(-numerator, denominator);
    }

    /**
     * <p>
     * Gets a fraction that is the positive equivalent of this one.
     * </p>
     * <p>
     * More precisely: <code>(fraction &gt;= 0 ? this : -fraction)</code>
     * </p>
     * 
     * <p>
     * The returned fraction is not reduced.
     * </p>
     * 
     * @return <code>this</code> if it is positive, or a new positive fraction instance with the opposite signed
     *         numerator
     */
    public Fraction abs() {
        if (numerator >= 0) {
            return this;
        }
        return negate();
    }

    /**
     * <p>
     * Gets a fraction that is raised to the passed in power.
     * </p>
     * 
     * <p>
     * The returned fraction is in reduced form.
     * </p>
     * 
     * @param power
     *            the power to raise the fraction to
     * @return <code>this</code> if the power is one, <code>ONE</code> if the power is zero (even if the fraction equals
     *         ZERO) or a new fraction instance raised to the appropriate power
     * @throws ArithmeticException
     *             if the resulting numerator or denominator exceeds <code>Integer.MAX_VALUE</code>
     */
    public Fraction pow(final int power) {
        if (power == 1) {
            return this;
        } else if (power == 0) {
            return ONE;
        } else if (power < 0) {
            if (power == Integer.MIN_VALUE) { // MIN_VALUE can't be negated.
                return this.invert().pow(2).pow(-(power / 2));
            }
            return this.invert().pow(-power);
        } else {
            final Fraction f = this.multipliedBy(this);
            if (power % 2 == 0) { // if even...
                return f.pow(power / 2);
            } else { // if odd...
                return f.pow(power / 2).multipliedBy(this);
            }
        }
    }

    /**
     * <p>
     * Gets the greatest common divisor of the absolute value of two numbers, using the "binary gcd" method which avoids
     * division and modulo operations. See Knuth 4.5.2 algorithm B. This algorithm is due to Josef Stein (1961).
     * </p>
     * 
     * @param u
     *            a non-zero number
     * @param v
     *            a non-zero number
     * @return
     */
    private static int greatestCommonDivisor(int u, int v) {
        // From Commons Math:
        if (u == 0 || v == 0) {
            if (u == Integer.MIN_VALUE || v == Integer.MIN_VALUE) {
                throw new ArithmeticException("overflow: gcd is 2^31");
            }
            return Math.abs(u) + Math.abs(v);
        }
        //if either operand is abs 1, return 1:
        if (Math.abs(u) == 1 || Math.abs(v) == 1) {
            return 1;
        }
        // keep u and v negative, as negative integers range down to
        // -2^31, while positive numbers can only be as large as 2^31-1
        // (i.e. we can't necessarily negate a negative number without
        // overflow)
        if (u > 0) {
            u = -u;
        } // make u negative
        if (v > 0) {
            v = -v;
        } // make v negative
          // B1. [Find power of 2]
        int k = 0;
        while ((u & 1) == 0 && (v & 1) == 0 && k < 31) { // while u and v are both even...
            u /= 2;
            v /= 2;
            k++; // cast out twos.
        }
        if (k == 31) {
            throw new ArithmeticException("overflow: gcd is 2^31");
        }
        // B2. Initialize: u and v have been divided by 2^k and at least
        //     one is odd.
        int t = (u & 1) == 1 ? v : -(u / 2)/* B3 */;
        // t negative: u was odd, v may be even (t replaces v)
        // t positive: u was even, v is odd (t replaces u)
        do {
            /* assert u<0 && v<0; */
            // B4/B3: cast out twos from t.
            while ((t & 1) == 0) { // while t is even..
                t /= 2; // cast out twos
            }
            // B5 [reset max(u,v)]
            if (t > 0) {
                u = -t;
            } else {
                v = t;
            }
            // B6/B3. at this point both u and v should be odd.
            t = (v - u) / 2;
            // |u| larger: t positive (replace u)
            // |v| larger: t negative (replace v)
        } while (t != 0);
        return -u * (1 << k); // gcd is u*2^k
    }

    // Arithmetic
    //-------------------------------------------------------------------

    /**
     * Multiply two integers, checking for overflow.
     * 
     * @param x
     *            a factor
     * @param y
     *            a factor
     * @return
     * @throws ArithmeticException
     *             if the result can not be represented as an int
     */
    private static int mulAndCheck(final int x, final int y) {
        final long m = (long) x * (long) y;
        if (m < Integer.MIN_VALUE || m > Integer.MAX_VALUE) {
            throw new ArithmeticException("overflow: mul");
        }
        return (int) m;
    }

    /**
     * Multiply two non-negative integers, checking for overflow.
     * 
     * @param x
     *            a non-negative factor
     * @param y
     *            a non-negative factor
     * @return
     * @throws ArithmeticException
     *             if the result can not be represented as an int
     */
    private static int mulPosAndCheck(final int x, final int y) {
        /* assert x>=0 && y>=0; */
        final long m = (long) x * (long) y;
        if (m > Integer.MAX_VALUE) {
            throw new ArithmeticException("overflow: mulPos");
        }
        return (int) m;
    }

    /**
     * Add two integers, checking for overflow.
     * 
     * @param x
     *            an addend
     * @param y
     *            an addend
     * @return
     * @throws ArithmeticException
     *             if the result can not be represented as an int
     */
    private static int addAndCheck(final int x, final int y) {
        final long s = (long) x + (long) y;
        if (s < Integer.MIN_VALUE || s > Integer.MAX_VALUE) {
            throw new ArithmeticException("overflow: add");
        }
        return (int) s;
    }

    /**
     * Subtract two integers, checking for overflow.
     * 
     * @param x
     *            the minuend
     * @param y
     *            the subtrahend
     * @return
     * @throws ArithmeticException
     *             if the result can not be represented as an int
     */
    private static int subAndCheck(final int x, final int y) {
        final long s = (long) x - (long) y;
        if (s < Integer.MIN_VALUE || s > Integer.MAX_VALUE) {
            throw new ArithmeticException("overflow: add");
        }
        return (int) s;
    }

    /**
     * <p>
     * Adds the value of this fraction to another, returning the result in reduced form. The algorithm follows Knuth,
     * 4.5.1.
     * </p>
     * 
     * @param fraction
     *            the fraction to add, must not be <code>null</code>
     * @return a <code>Fraction</code> instance with the resulting values
     * @throws IllegalArgumentException
     *             if the fraction is <code>null</code>
     * @throws ArithmeticException
     *             if the resulting numerator or denominator exceeds <code>Integer.MAX_VALUE</code>
     */
    public Fraction add(final Fraction fraction) {
        return addSub(fraction, true /* add */);
    }

    /**
     * <p>
     * Subtracts the value of another fraction from the value of this one, returning the result in reduced form.
     * </p>
     * 
     * @param fraction
     *            the fraction to subtract, must not be <code>null</code>
     * @return a <code>Fraction</code> instance with the resulting values
     * @throws IllegalArgumentException
     *             if the fraction is <code>null</code>
     * @throws ArithmeticException
     *             if the resulting numerator or denominator cannot be represented in an <code>int</code>.
     */
    public Fraction subtract(final Fraction fraction) {
        return addSub(fraction, false /* subtract */);
    }

    /**
     * Implement add and subtract using algorithm described in Knuth 4.5.1.
     * 
     * @param fraction
     *            the fraction to subtract, must not be <code>null</code>
     * @param isAdd
     *            true to add, false to subtract
     * @return a <code>Fraction</code> instance with the resulting values
     * @throws IllegalArgumentException
     *             if the fraction is <code>null</code>
     * @throws ArithmeticException
     *             if the resulting numerator or denominator cannot be represented in an <code>int</code>.
     */
    private Fraction addSub(final Fraction fraction, final boolean isAdd) {
        if (fraction == null) {
            throw new IllegalArgumentException("The fraction must not be null");
        }
        // zero is identity for addition.
        if (numerator == 0) {
            return isAdd ? fraction : fraction.negate();
        }
        if (fraction.numerator == 0) {
            return this;
        }
        // if denominators are randomly distributed, d1 will be 1 about 61%
        // of the time.
        final int d1 = greatestCommonDivisor(denominator, fraction.denominator);
        if (d1 == 1) {
            // result is ( (u*v' +/- u'v) / u'v')
            final int uvp = mulAndCheck(numerator, fraction.denominator);
            final int upv = mulAndCheck(fraction.numerator, denominator);
            return new Fraction(isAdd ? addAndCheck(uvp, upv) : subAndCheck(uvp, upv), mulPosAndCheck(denominator, fraction.denominator));
        }
        // the quantity 't' requires 65 bits of precision; see knuth 4.5.1
        // exercise 7.  we're going to use a BigInteger.
        // t = u(v'/d1) +/- v(u'/d1)
        final BigInteger uvp = BigInteger.valueOf(numerator).multiply(BigInteger.valueOf(fraction.denominator / d1));
        final BigInteger upv = BigInteger.valueOf(fraction.numerator).multiply(BigInteger.valueOf(denominator / d1));
        final BigInteger t = isAdd ? uvp.add(upv) : uvp.subtract(upv);
        // but d2 doesn't need extra precision because
        // d2 = gcd(t,d1) = gcd(t mod d1, d1)
        final int tmodd1 = t.mod(BigInteger.valueOf(d1)).intValue();
        final int d2 = tmodd1 == 0 ? d1 : greatestCommonDivisor(tmodd1, d1);

        // result is (t/d2) / (u'/d1)(v'/d2)
        final BigInteger w = t.divide(BigInteger.valueOf(d2));
        if (w.bitLength() > 31) {
            throw new ArithmeticException("overflow: numerator too large after multiply");
        }
        return new Fraction(w.intValue(), mulPosAndCheck(denominator / d1, fraction.denominator / d2));
    }

    /**
     * <p>
     * Multiplies the value of this fraction by another, returning the result in reduced form.
     * </p>
     * 
     * @param fraction
     *            the fraction to multiply by, must not be <code>null</code>
     * @return a <code>Fraction</code> instance with the resulting values
     * @throws IllegalArgumentException
     *             if the fraction is <code>null</code>
     * @throws ArithmeticException
     *             if the resulting numerator or denominator exceeds <code>Integer.MAX_VALUE</code>
     */
    public Fraction multipliedBy(final Fraction fraction) {
        if (fraction == null) {
            throw new IllegalArgumentException("The fraction must not be null");
        }
        if (numerator == 0 || fraction.numerator == 0) {
            return ZERO;
        }
        // knuth 4.5.1
        // make sure we don't overflow unless the result *must* overflow.
        final int d1 = greatestCommonDivisor(numerator, fraction.denominator);
        final int d2 = greatestCommonDivisor(fraction.numerator, denominator);
        return of(mulAndCheck(numerator / d1, fraction.numerator / d2), mulPosAndCheck(denominator / d2, fraction.denominator / d1), true);
    }

    /**
     * <p>
     * Divide the value of this fraction by another.
     * </p>
     *
     * @param fraction the fraction to divide by, must not be <code>null</code>
     * @return a <code>Fraction</code> instance with the resulting values
     * @throws IllegalArgumentException             if the fraction is <code>null</code>
     * @throws ArithmeticException             if the resulting numerator or denominator exceeds <code>Integer.MAX_VALUE</code>
     */
    public Fraction dividedBy(final Fraction fraction) {
        if (fraction == null) {
            throw new IllegalArgumentException("The fraction must not be null");
        }
        if (fraction.numerator == 0) {
            throw new ArithmeticException("The fraction to divide by must not be zero");
        }
        return multipliedBy(fraction.invert());
    }

    // Basics
    //-------------------------------------------------------------------

    /**
     * <p>
     * Compares this object to another based on size.
     * </p>
     * 
     * <p>
     * Note: this class has a natural ordering that is inconsistent with equals, because, for example, equals treats 1/2
     * and 2/4 as different, whereas compareTo treats them as equal.
     * 
     * @param other
     *            the object to compare to
     * @return -1 if this is less, 0 if equal, +1 if greater
     * @throws ClassCastException
     *             if the object is not a <code>Fraction</code>
     * @throws NullPointerException
     *             if the object is <code>null</code>
     */
    @Override
    public int compareTo(final Fraction other) {
        if (this == other) {
            return 0;
        }
        if (numerator == other.numerator && denominator == other.denominator) {
            return 0;
        }

        // otherwise see which is less
        final long first = (long) numerator * (long) other.denominator;
        final long second = (long) other.numerator * (long) denominator;
        if (first == second) {
            return 0;
        } else if (first < second) {
            return -1;
        } else {
            return 1;
        }
    }

    /**
     * <p>
     * Gets the fraction as a proper <code>String</code> in the format X Y/Z.
     * </p>
     * 
     * <p>
     * The format used in '<i>wholeNumber</i> <i>numerator</i>/<i>denominator</i>'. If the whole number is zero it will
     * be omitted. If the numerator is zero, only the whole number is returned.
     * </p>
     * 
     * @return a <code>String</code> form of the fraction
     */
    public String toProperString() {
        if (toProperString == null) {
            if (numerator == 0) {
                toProperString = "0";
            } else if (numerator == denominator) {
                toProperString = "1";
            } else if (numerator == -1 * denominator) {
                toProperString = "-1";
            } else if ((numerator > 0 ? -numerator : numerator) < -denominator) {
                // note that we do the magnitude comparison test above with
                // NEGATIVE (not positive) numbers, since negative numbers
                // have a larger range.  otherwise numerator==Integer.MIN_VALUE
                // is handled incorrectly.
                final int properNumerator = getProperNumerator();
                if (properNumerator == 0) {
                    toProperString = Integer.toString(getProperWhole());
                } else {
                    toProperString = new StringBuilder(32).append(getProperWhole())
                            .append(' ')
                            .append(properNumerator)
                            .append('/')
                            .append(getDenominator())
                            .toString();
                }
            } else {
                toProperString = new StringBuilder(32).append(getNumerator()).append('/').append(getDenominator()).toString();
            }
        }
        return toProperString;
    }

    /**
     * <p>
     * Compares this fraction to another object to test if they are equal.
     * </p>
     * .
     * 
     * <p>
     * To be equal, both values must be equal. Thus 2/4 is not equal to 1/2.
     * </p>
     * 
     * @param obj
     *            the reference object with which to compare
     * @return <code>true</code> if this object is equal
     */
    @Override
    public boolean equals(final Object obj) {
        if (obj == this) {
            return true;
        }
        if (obj instanceof Fraction == false) {
            return false;
        }
        final Fraction other = (Fraction) obj;
        return getNumerator() == other.getNumerator() && getDenominator() == other.getDenominator();
    }

    /**
     * <p>
     * Gets a hashCode for the fraction.
     * </p>
     * 
     * @return a hash code value for this object
     */
    @Override
    public int hashCode() {
        if (hashCode == 0) {
            // hashcode update should be atomic.
            hashCode = 37 * (37 * 17 + getNumerator()) + getDenominator();
        }
        return hashCode;
    }

    /**
     * <p>
     * Gets the fraction as a <code>String</code>.
     * </p>
     * 
     * <p>
     * The format used is '<i>numerator</i>/<i>denominator</i>' always.
     * 
     * @return a <code>String</code> form of the fraction
     */
    @Override
    public String toString() {
        if (toString == null) {
            toString = new StringBuilder(32).append(getNumerator()).append('/').append(getDenominator()).toString();
        }
        return toString;
    }
}
